We present a parallel algorithm for computing the minimum s-t cut in structured 3-dimensional proper order graphs arising from image segmentation problems. Proper order graphs are multi-column structures where vertices are arranged in parallel columns, with each vertex connected to consecutive vertices in adjacent columns. This graph structure naturally arises in surface extraction problems for geological horizon segmentation in seismic imaging volumes. We develop two parallel approaches: a hierarchical merging variant of the Boykov-Kolmogorov algorithm, and a novel parallel push-relabel algorithm with level synchronized global relabeling. Our primary contribution is the push-relabel variant, which partitions the graph into segments along columns with processor affinity, eliminating the need for a global shared queue. We introduce level synchronized global relabeling that enables concurrent label updates while maintaining correctness through barriers at each frontier level.

翻译：本文提出一种并行算法，用于计算图像分割问题中结构化三维正序图的最小s-t割。正序图是一种多列结构，其中顶点按平行列排列，每个顶点与相邻列中的连续顶点相连。该图结构天然出现在地震成像体地质层位分割的表面提取问题中。我们开发了两种并行方法：Boykov-Kolmogorov算法的分层合并变体，以及一种具有层级同步全局重标记的新型并行推送-重标记算法。我们的主要贡献在于推送-重标记变体，该算法沿列方向将图分割为具有处理器亲和性的区段，从而消除了对全局共享队列的需求。我们提出的层级同步全局重标记机制，允许在保持正确性的前提下通过各前沿层级的屏障实现并发标记更新。